Individuals today may allocate their investment resources among a variety of asset classes; for example, equity, fixed income, international, emerging markets, etc. Within each asset class are a great number of individual assets to analyze and select. Investors may diversify and obtain professional management of their investment resources by investing in a professionally managed mutual fund. However, there are literally thousands of mutual funds to choose from offering a bewildering array of different investment philosophies. Many individuals do not have the time, inclination or expertise to manage their investments optimally or even choose the best mutual fund for their investment goals. Optimal investment of resources among a variety of assets is a complicated statistical and computationally intensive process beyond the reach of most investors.
Ideally an investor should allocate his or her investment to achieve a maximum expected rate of investment return consistent with the investor's tolerance for risk. A portfolio that is suitable for a particular investor can be constructed by combining assets with different expected rates of return and different levels of risk.
The expected rate of return for a combined portfolio of assets with different expected rates of return is the sum of the expected rates of return of each individual asset in the portfolio weighted by its proportion to the total portfolio: ##EQU1##
where:
r.sub.T is the expected rate of return of the portfolio of combined assets; PA1 r.sub.i is the expected rate of return of the i.sup.th asset; PA1 w.sub.i is the proportion of the value of the i.sup.th asset to the total portfolio value, ##EQU2## PA1 N is the total number of assets in the portfolio. PA1 r is a random variable representing the rate of return on an asset or portfolio; PA1 r is the expected value of r; and PA1 E denotes the expectation operator.
For example, consider a stock, A, with an expected rate of return r.sub.1 =0.1 and a bond, B, with an expected rate of return r=0.05. The expected rate of return of a portfolio consisting of 40% stock A and 60% bond B will be: EQU r.sub.T =0.1.times.0.4+0.05.times.0.6=0.07
Risk may be characterized in different ways. Probably the most common measure of risk is volatility, measured by standard deviation. Standard deviation is the square root of the variance of the returns of an asset or portfolio of assets. The variance is a measure of the extent to which the return on an asset or portfolio of assets deviates from an expected return. An asset with a higher standard deviation will be considered more risky than an asset with a lower standard deviation. Other measures of risk include semi-variance about a target return, which is a measure of the extent to which the return of an asset or portfolio of assets will fall below a target level of return. Another measure of risk is "value at risk," which is a measure of how much an asset or portfolio of assets can lose in value with a given probability.
The risk level of a combined portfolio of assets will depend on the risk measure used. For example, consider the risk associated with a combined portfolio using variance, or equivalently, standard deviation as the measure of risk. The standard deviation of the returns of a risk-free asset is zero whereas the standard deviation of the returns of a risky asset is greater than zero. Standard deviation is the square root of the variance. The variance is: EQU E{(r-r).sup.2 }
where:
Combining a plurality of risky and risk-free assets in a portfolio will result in a portfolio with a standard deviation that is equal to or less than the weighted sum of the standard deviations of the component assets. For example, when two risky assets with variances .sigma..sub.1.sup.2 and .sigma..sub.2.sup.2, respectively, are combined into a portfolio with portfolio weights w.sub.1, and w.sub.2, respectively, the portfolio variance, .sigma..sub.T.sup.2, is given by: ##EQU3##
where cov (r.sub.1, r.sub.2) =the covariance of the two assets.
The covariance is a measure of how much the returns on the two assets move in tandem, and is defined as follows: EQU cov (r.sub.1,R.sub.2)=.sigma..sub.12 =E[(r.sub.1 -r.sub.1)(r.sub.2 -r.sub.2)]
A positive covariance means that the asset returns move together; if one has a positive deviation from its mean, they both do. A negative covariance means that asset returns move in opposite directions; if one has a positive deviation from its mean, the other has a negative deviation from its mean. The correlation coefficient, .rho..sub.12, is the covariance of the two assets divided by the product of their standard deviations (i.e., .rho..sub.12 =.sigma..sub.12 /(.sigma..sub.1.sigma..sub.2)). The correlation coefficient, .rho..sub.12 may range from -1 (indicating perfect negative correlation) and +1 (indicating perfect positive correlation). Thus, the magnitude of the correlation coefficient, .vertline..rho..sub.12.vertline., is always less than or equal to 1.
The equation for the variance of the portfolio, .sigma..sub.T.sup.2, shows that a positive covariance increases portfolio variance beyond .SIGMA. w.sub.i.sup.2 .sigma..sub.i.sup.2. A negative covariance decreases portfolio variance. By investing in two assets that are negatively correlated, if one asset has a return greater than its expected return, that positive deviation should be offset by the extent to which the return of the other asset falls below its expected return.
The equation for .sigma..sub.T further shows that the standard deviation of the portfolio is always equal to (in the case .rho..sub.12 =1) or less than (in the case .vertline..rho..sub.12.vertline.&lt;1) the weighted sum of the standard deviations of the component assets. That is: EQU .sigma..sub.T.ltoreq.w.sub.1.sigma..sub.1 +w.sub.2.sigma..sub.2
Since the return of the combined portfolio is the weighted average of the returns of the component assets, portfolios of less-than-perfectly correlated assets always offer better risk return opportunities than the individual component securities. See, e.g., "Investments, 3rd Edition," p. 197, Bodie, Kane & Marcus, Irwin, McGraw Hill (1996). These results are true generally for a combined portfolio comprising numerous risky assets, for which the variance is given by: ##EQU4##
Thus, since the magnitude of the correlation coefficient, .vertline..rho..sub.12.vertline., for any two different assets, (a.sub.i, a.sub.j), is less than or equal to 1, .sigma..sub.T.sup.2 is always less than or equal to ##EQU5##
Thus, .sigma..sub.T is always less than or equal to ##EQU6##
Given a set of imperfectly correlated risky assets, an innumerable set of combined portfolios can be constructed, each comprising different proportions of the component assets. An optimum portfolio is one in which the proportion of each asset comprising the portfolio results in the highest expected return for the combined portfolio for a given level of risk. Alternatively, an optimum portfolio is one in which the proportion of each asset comprising the portfolio minimizes the risk of the combined portfolio for any targeted expected return. See, e.g., "Investments, 3rd Edition," Bodie, Kane & Marcus, Irwin, McGraw Hill (1996).
This is illustrated in FIG. 1, using variance, or equivalently, standard deviation, as the risk measure. FIG. 1 is a graph of the minimum variance frontier of risky assets. This frontier is a graph of the expected risk and return of the portfolios with the lowest possible risk for given expected returns. This graph can be obtained by finding the set of weights for each component asset that will give the minimum variance for each targeted expected return.
The global minimum variance, Point A in FIG. 1, is the lowest variance that can be achieved, given the assets selected to comprise the portfolio. The portion of the minimum variance frontier that is concave downward, lying above and to the right of the global minimum variance, is called the efficient frontier. In FIG. 1, the efficient frontier is represented by the solid line above and to the right of Point A. The portion of the curve that is concave up from the global minimum variance represents inefficient points (portfolios), as there are points that lie directly above with higher expected return at the same level of risk (those points on the efficient frontier). In FIG. 1, these points are represented by the dashed line below and to the right of Point A.
In FIG. 2, the shaded area represents where the efficient frontier will always lie. Points A and B in FIG. 2 represent two portfolios on the efficient frontier. If the portfolios represented by Points A and B are perfectly positively correlated (.rho.=1), the efficient frontier curve is the solid line connecting Points A and B. This line is the weighted average of any combination of the two portfolios. If the portfolios represented by Points A and B are perfectly negatively correlated (.rho.-1), the efficient frontier curve is the dashed lines connecting Points A and B. This line shows that a certain combined portfolio of A and B will have a risk level equal to zero. For any portfolios A and B which are not perfectly correlated (-1&lt;.rho.&lt;1), the efficient frontier curve must lie in the shaded area of FIG. 2, which is bordered by the perfectly positive and negative correlation lines. To lie in this shaded area between any two points on the efficient frontier, it must have the concave downward shape.
The efficient frontier is the curve that yields the highest expected return for a given level of risk. All other combinations of the assets selected to comprise the portfolio will result in a lower expected return for a given risk level. In particular, the risk-return plot of each individual asset will lie below and to the right of the efficient frontier. The efficient frontier represents the optimum risk-return opportunities available to an investor from a portfolio of risky assets.
Although variance is the measure of risk used to depict the efficient frontier in FIG. 1, an efficient frontier may be determined in terms of other risk measures as well. (See for example, "Post-Modern Portfolio Theory Comes of Age," B. Rom and K. Ferguson, The Journal of Investing, Fall 1994).
Ideally, a rational investor would choose to invest in a portfolio corresponding to the point on the efficient frontier that yields the highest expected return consistent with the investor's tolerance for risk. An investor who is highly risk averse should choose a point on the efficient frontier that provides a lower risk than an investor who is less risk averse. Consequently, the investor that is more highly risk averse will attain a lower expected return than would be attained by the less risk averse investor. Nevertheless, by choosing a portfolio that lies on the efficient frontier, each investor will attain the highest expected return attainable for a given level of risk. In practice, however, most investors lack the time, knowledge or inclination to perform the calculations required to construct a portfolio on an efficient frontier.
Many investment products available today offer the investor various risk-return choices. For example, an investor may choose among a finite set of portfolios, each comprising a different preselected mix of assets corresponding to a different risk preference. These products provide separate portfolios for each of a set of different risk preferences that may not lie on the efficient frontier. Similarly, investment products which allow an investor to select his or her own mix of various assets or portfolios will generally not result in a portfolio that lies on the efficient frontier. In short, investment products currently available to investors are suboptimal. They fail to provide the investor with the highest attainable expected return for a given level of risk. Also, current products do not pool investors with different risk tolerances to take advantage of diversification.
Therefore, what is needed is a system and method for allocating the returns from a single portfolio to a plurality of investors with different risk tolerances as a function of the preferred risk-return combinations chosen by the investors. Also, what is needed is a system that will provide to each investor risk-return opportunities that lie on or above an efficient frontier so that each investor will attain the highest achievable expected return for a given level of risk and the potential to earn more than the investor could have earned if the investor invested in an efficient portfolio on his or her own.